Theorem. Let $R$ be a domain, $K$ its fraction field and $L$ a field extension of $K$. Let $S$ be the integral closure of $R$ in $L$. Let $P_1,...,P_k$ be prime ideals in $S$ with $P_i \cap R = p$. Then there is always a solution $x$ to systems of the form $x \equiv x_i \bmod P_i$.
For a maximal ideal I see how CRT solves this. The lecture notes claim that we may assume $p$ is maximal by localizing at $p$. How is this reduction done?
Is it clear that $P_i$ stay distinct upon localizing at $M =R \setminus p$?
Given target $x_i$, how do we raise some solution $x_p \in S_M$ for $x_p - y_i \in M^{-1}P_i$ to a solution in $S$?
The notes mention an isomorphism $S/pS \cong S_M/pS_M$. Is it relevant?